Homework $1.$ Quantitative Methods for fixed Income Securities

CHAPTER $1$ Bond Prices, Discount Factors, and Arbitrage $(Tuckman)$

For the following problems, $we$ assume that today $is$ May $15,2024.$

$1\mathrm{.}1.(4)$ Write down the cash flow dates and the cash flows $of$ $$1,000$ face value $of$ the $U.S.$ Treasury $4sof$ April $30,2026,$ issued $on$ April $30,2024.$

$1\mathrm{.}2(4)$ The Treasury $5sof$ February $15,2033,$ which was issued $on$ February $15,2023,is$ purchased $on$ May $15,2024,$ for a quoted price $of96-\frac{231}{2.}$ What $is$ the invoice price $(i.e.$ full price$)on$ $$100,000$ face amount?

$1\mathrm{.}3(6)$ Here $is$ a list $of$ bond transactions $on$ May $15,2024.$ For each transaction find out the transaction price.

Bond

$10\mathrm{.}75sof\frac{5}{15}\frac{?}{2026}4\mathrm{.}25sof\frac{11}{15}\frac{?}{2026}7\mathrm{.}25sof\frac{5}{15}\frac{?}{2027}$

Price Face Amount

$112-\frac{25}{8}$ $$10,00099-14+$ $$1,000107-4$ $$1,000,000$

$1\mathrm{.}4(6)$ Use the Treasury bond prices $asof$ May $15,2024$ listed below $to$ derive the discount factors for cash flows $tobe$ received $in6$ months, $1$ year, and $1\mathrm{.}5$ years.

Bond

$7\mathrm{.}5sof\frac{11}{15}\frac{?}{2024}7\mathrm{.}5sof\frac{5}{15}\frac{?}{2025}11\mathrm{.}625sof\frac{11}{15}\frac{?}{2025}$

Price

$101-\frac{253}{4}103-\frac{1215}{16}110-\frac{211}{4}$

$1\mathrm{.}5(4)$ Suppose there exists a Treasury issue with $a7\mathrm{.}5\%$ coupon maturing $on$

November $15,2025.$ According $to$ the discount factors derived $in$ question $1\mathrm{.}4,$

what would $be$ the price $of$ the $7\mathrm{.}5sof$ November $15,2025?$

$1\mathrm{.}6(8)[Continued$ from $1\mathrm{.}5]$ Say that the $7\mathrm{.}5sof$ November $15,2025,$ existed and traded $at$ a price $of105$ instead $of$ the price derived $in$ question $1\mathrm{.}5.$ How could one earn $an$ arbitrage profit $by$ trading the $7\mathrm{.}5sof$ November $15,2025,$ and the three bonds listed $in$ question $1\mathrm{.}4?$ Using the prices listed $in$ question $1\mathrm{.}4,$ how much arbitrage profit $is$ available $in$ this trade?